The following question is taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
Exercise: Let $\textbf{K}$ be any category and let $\textbf{C}$ be any subset of $Obj(\textbf{K}).$ Show that $Obj(\bar{\textbf{C}})=\textbf{C},$ $\bar{\textbf{C}}(A,B)=\textbf{K}(A,B)$ (for $A,B,\in \textbf{C},$) with composition and identities as in $\textbf{K},$ defines a category $\bar{\textbf{C}},$ the $\textbf{full subcategory induced by C}.$
Partial solution:
Since $\textbf{C}\subset Obj(\textbf{K})$, then we can consider the subset $\textbf{C}$ to be an object in the category of $\textbf{K}$ and also to be equal to $Obj(\bar{\textbf{C}})$. Since $\textbf{C}\subset Obj(\textbf{K})$, then for any $A,B\in \textbf{C}=Obj(\bar{\textbf{C}}),$ The set of morphisms $\bar{\textbf{C}}(A,B)\subset \textbf{K}(A,B).$ From here, I don't know how to show $\textbf{K}(A,B)\subset \bar{\textbf{C}}(A,B).$ I need to show the reverse conclusion since for the case of composition also requires it. Thank you in advance.
You're thinking too complicated: The assignment does not require you to show that $\mathbf{\overline C}(A, B)=\mathbf{K}(A, B)$, but rather gives it as a definition. All that's left to show is that this data (Object set and homsets, together with identities and composition inherited by $\mathbf{K}$), still satisfies the requirements of a category (identities behaving as identities and associativity of composition).